CN109459930A - A kind of cooperative control method based on PD structure and neighbours' Delay control signal - Google Patents
A kind of cooperative control method based on PD structure and neighbours' Delay control signal Download PDFInfo
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Abstract
The present invention is directed to the trajectory synchronization tracking problem of second order multiple agent, propose a kind of cooperative control method based on PD structure and neighbours' Delay control signal, the invention belongs to Collaborative Control technical fields, main proportional-plus-derivative (PD) control that information (CIIN) and local neighborhood synchronous error signal (LNSE) are inputted using last time neighbours, to construct controller.This controller design method has control precision high, and communication cost is small, the continuous advantage of control signal.Insight of the invention is that firstly, building tracking error signal and local neighborhood synchronous error signal.Secondly, controlling input information and local neighborhood synchronous error signal using last time neighbours, a kind of distributed director is designed.Finally, determining controller parameter according to an inequality condition, the ratio for guaranteeing system stability and differential control gain condition are given, form is succinct, is easy to calculate and verify.
Description
Technical field
The invention belongs to Collaborative Control technical fields, mainly for the trajectory synchronization tracking problem of second order multi-agent system
Propose a kind of simple distributed director design method of structure, mainly using last time neighbours input information (CIIN) and
The proportional-plus-derivative (PD) of local neighborhood synchronous error signal (LNSE) controls, to construct controller.
Background technique
Multi-agent system, can be by mutual between intelligent body by the system of multiple interactive intelligent main bodys formed
Communication is coordinated to solve large-scale and complex realistic problem.Since the Collaborative Control of multi-agent system is compared to single individual
With better robustness, flexibility and economy, multiple agent cooperate with tracing control research in space exploration, mobile robot
Cooperation, formation control, vehicle are unanimously dispatched etc. field and are all widely used.
In practical projects, multi-agent system is generally larger, traditional due to the limitation of sensor and communication
Centralized control mode is no longer desirable for these scenes, has and calculates small cost, strong real-time, robustness and zmodem, leads to
Letter requires relatively low, and distributed AC servo system strategy the advantages that system flexible design is widely used.It is existing many for more
The distributed AC servo system strategy of multiagent system trajectory synchronization tracking problem, such as sliding mode controller, adaptive controller, output are adjusted
Section controller all haves the defects that certain.Sliding mode controller can constantly become in dynamic process according to system current state
Change, system forced to move according to the track of predetermined " sliding mode ", however this control input be it is discontinuous, to practical application
Cause certain difficulty.Adaptive controller needs constantly to extract model sophisticated model for information about using parameter estimator, defeated
Adjusting controller needs to estimate using progressive observer external reference signal and interference out, and both the above control strategy structure is all
It is complex.And proportional-integral-differential (PID) controller and its deformation are widely used in intelligence due to its succinct validity
Body control.
Summary of the invention
The present invention is proposed and a kind of controls input information using last time neighbours to overcome the defect of above controller
(CIIN) it is communicated with the method for local neighborhood synchronous error signal (LNSE) proportional-plus-derivative (PD) control with control precision height
Cost is small, the continuous advantage of control signal.Insight of the invention is that firstly, building tracking error signal is synchronous with local neighborhood
Error signal.Secondly, controlling input information and local neighborhood synchronous error signal using last time neighbours, a kind of distribution is designed
Formula controller.Finally, controller parameter is determined according to an inequality condition, to guarantee that multi-agent system trajectory synchronization tracks
Effect meet the requirements.
Technical solution of the present invention is a kind of cooperative control method based on PD structure and neighbours' Delay control signal, this method
Include:
Step 1: building tracking error signal, local neighborhood synchronous error signal, communication topology figure；
For the system for having n second order multiple agent:
Wherein pi(t)、qi(t)、ui(t) position, the speed, control input of i-th of intelligent body are respectively indicated,The position of i-th of intelligent body, the derivative of speed are respectively indicated, i.e.,Indicate the acceleration of i-th of intelligent body
Degree；
The traffic model of multi-agent system is established using algebraic graph theory；
For the multi-agent system comprising n follower and 1 pilotage people, regard intelligent body each in system as one
Point, the information exchange between intelligent body regards side between points as, with a digraph or non-directed graphTo carve
The Communication topology of multi-agent system is drawn, whereinFor point set, v1,v2,…,vnFor n point,Side collection；If A=[aij]∈Rn×nFor figureAdjacency matrix, Rn×nIt indicates that n × n ties up real linear space, uses
aijIndicate the connected relation between node, and if only ifWhen aij=1, otherwise aij=0, if aii=0；Definition
Indegree matrix D=diag (d1, d2,…dn), whereinNiIndicate the set of node i neighbours；It is asked considering to track
When topic, figure is usedExpression contains the system communication topological diagram of n follower and a pilotage people, uses matrix B=diag
(b1,b2,…bn) come describe the information between pilotage people and follower transmitting situation, if i-th of follower can receive navigator
The information of person, then bi=1, otherwise bi=0；Use μiRepresenting matrix A (D+B)-1Characteristic value, i=1,2 ... n；
The kinetic model of frame of reference are as follows:
Wherein: p0(t), q0(t) desired locations and speed of frame of reference generation are respectively indicated,Respectively
Indicate the position of frame of reference, the derivative of speed, i.e.,Indicate the acceleration of frame of reference, u0(t) frame of reference is indicated
Input signal；
Tracking error signal are as follows:
Wherein:Indicate the position of i-th of intelligent body and the error of desired locations,Indicate i-th of intelligent body
Speed and desired speed error；
Local neighborhood synchronous error signal:
Wherein: epi(t) position synchronous error, e are indicatedqi(t) speed synchronous error is indicated；
The communication topology figure of configuration system, and meet following two condition: 1)There are directed spanning trees；2) matrix A (D+
B)-1N all characteristic value μiIt is all real number；
Step 2: inputting information and local neighborhood synchronous error signal using the control of last time neighbours, design is distributed
Controller；
Control target is that the position of each intelligent body i can track the position of pilotage people while realize mutual
Synchronous, the neighbours based on intelligent body i control input u thusjWith synchronous error signal epiAnd eqi, design distributed AC servo system rule are as follows:
Whereinui(t) the control input of i-th of intelligent body, u are indicatedj(t) i-th of intelligence is indicated
The control input of energy body, kP> 0 is ratio term coefficient, kD> 0 is differential term coefficient；In order to avoid there is generation when control law resolving
Ring of numbers problem controls input u in neighboursjIn introduce fixed delay τ, the improved form for obtaining control law is as follows:
Step 3: when guaranteeing that closed-loop system is stablized, design parameter kPAnd kDThe condition that should meet, to guarantee multi-agent system
The effect of trajectory synchronization tracking is met the requirements；
DefinitionckIt is necessary to meet following condition:
Determine design parameter kPAnd kD, so that condition (7) meets, then to any delay, τ > 0, tracking errorThe stability of uniform ultimate bounded, i.e. multi-agent system is unrelated with delay, τ；In addition, if pilotage people
Acceleration u0(t) be the time a Lipchitz function,So τ is smaller,The final boundary of tracking error is smaller；Particularly,Such as
Fruit u0(t) meet；
As t → ∞, tracking errorAsymptotic convergence is to null vector；
Step 4: if distributed director parameter kP,kDIt is unsatisfactory for the upper bound that calculation delay τ allows when condition (7), when τ's
When practical value is less than the upper bound, multi-agent system is stablized；Specifically include following sub-step:
(1) it calculatesWith
(2) it solves equationObtain θ；
(3) it calculates
(4) maximum delay that can guarantee system stability is calculated
Step 5: Collaborative Control is carried out to multiple agent according to the result of step 2,3,4.
The present invention proposes a kind of distributed AC servo system rule for the trajectory synchronization tracking problem of second order multi-agent system,
Structure is relatively simple, is easily achieved in engineering；Give the ratio for guaranteeing system stability and differential control gain condition, shape
Formula is succinct, is easy to calculate and verify；Compared with existing some common distributed AC servo system rules, have control precision high, communication cost
It is small, the continuous advantage of control signal.
Detailed description of the invention
Fig. 1: undirected communication topology figure, r represent leader, and number 1,2,3,4 represents follower
Fig. 2: oriented communication topology figure, r represent leader, and number 1,2,3,4 represents follower
Fig. 3 a: to non-directed graph 1, parameter τ=0.01s, k are setP=10, kDEach intelligent body position tracking when=0.2 is missed
Difference.
Fig. 3 b: to digraph 2, parameter τ=0.01s, k are setP=10, kDEach intelligent body position tracking when=0.2 is missed
Difference.
Fig. 4 a: to non-directed graph 1, parameter τ=0.01s, k are setP=10, kDEach intelligent body location track when=12.
Fig. 4 b: to non-directed graph 1, parameter τ=0.01s, k are setP=10, kDEach intelligent body position tracking error when=12.
Fig. 5 a: to non-directed graph 1, parameter τ=0.01s, k are setP=1, kDEach intelligent body location track when=4.
Fig. 5 b: to non-directed graph 1, parameter τ=0.01s, k are setP=1, kDEach intelligent body position tracking error when=4.
Fig. 6 a: to non-directed graph 1, parameter τ=0.1s, k are setP=1, kDEach intelligent body location track when=4.
Fig. 6 b: to non-directed graph 1, parameter τ=0.1s, k are setP=1, kDEach intelligent body position tracking error when=4.
Fig. 7 a: to non-directed graph 1, parameter τ=10s, k are setP=1, kDEach intelligent body location track when=4.
Fig. 7 b: to non-directed graph 1, parameter τ=10s, k are setP=1, kDEach intelligent body position tracking error when=4.
Fig. 8 a: to digraph 2, parameter τ=0.01s, k are setP=10, kDEach intelligent body location track when=5.
Fig. 8 b: to digraph 2, parameter τ=0.01s, k are setP=10, kDEach intelligent body position tracking error when=5.
Fig. 9 a: to digraph 2, parameter τ=0.01s, k are setP=1, kDEach intelligent body location track when=2.
Fig. 9 b: to digraph 2, parameter τ=0.01s, k are setP=1, kDEach intelligent body position tracking error when=2.
Figure 10 a: to digraph 2, parameter τ=0.1s, k are setP=1, kDEach intelligent body location track when=2.
Figure 10 b: to digraph 2, parameter τ=0.1s, k are setP=1, kDEach intelligent body position tracking error when=2.
Figure 11 a: to digraph 2, parameter τ=10s, k are setP=1, kDEach intelligent body location track when=2.
Figure 11 b: to digraph 2, parameter τ=10s, k are setP=1, kDEach intelligent body position tracking error when=2.
Figure 12 a: to non-directed graph 1, parameter τ=0.01s, k are setP=1, kDEach intelligent body position tracking error when=6.
Figure 12 b: to non-directed graph 1, parameter τ=10s, k are setP=1, kDEach intelligent body position tracking error when=6.
Figure 13 a: to digraph 2, parameter τ=0.01s, k are setP=1, kDEach intelligent body position tracking error when=4.
Figure 13 b: to digraph 2, parameter τ=10s, k are setP=1, kDEach intelligent body position tracking error when=4.
Specific embodiment
Design object of the invention is to design a kind of trajectory synchronization tracking of distributed director solution multi-agent system
Problem.
In specific implementation, closed-loop control system experiment porch is built using the tool box Simulink in Matlab, comprising:
Multi-agent system, the i.e. kinetic model of the kinetic model of follower and pilotage people；Distributed director.Secondly, selection
Different control law parameter τ, kP,kD, in the case of comparative analysis different parameters, the tracking of each intelligent body and synchronous effect.
Its specific implementation step is as follows:
The first step calculates c according to the communication topological diagram of multi-agent systemk
First, it is assumed that then two kinds of Communication topologies calculate c as depicted in figs. 1 and 2kRange, obtain kP,kDIt needs
The condition to be met.
Second step sets desired signal according to actual scene
In simulations, it is as follows that two kinds of frames of reference are defined:
Third step builds close loop control circuit and builds closed-loop control system reality using the tool box Simulink in Matlab
Test platform.Multi-Agent System Model is built, here includes 4 follower, kinetic model such as navigates shown in (1) with 1
Person considers two kinds of kinetic models (10) (11) respectively.Then distributed director (6) are built, wherein delay, τ, ratio term system
Number kP, differential term coefficient kDIt is all adjustable parameter.
4th step selects different control law parameter τ, k to frame of reference (10)P,kD, each under comparative analysis different situations
The tracking effect of a intelligent body.
(1) k of ineligible (7) is selectedP,kD, observe the stability of closed-loop system
Selection parameter τ=0.01s, kP=10, kD=0.2, then ck=0.004, simulation result is as shown in Figure 3.It is logical to two kinds
Believe structure, even if the delay, τ very little of setting is 0.01s, but when the time tending to be infinite, position tracking error all dissipates, closed loop
System is unstable.The maximum delay that closed-loop system can be made stable at this time is calculated according to the step four in summary of the inventionIt obtains:
To non-directed graph 1,
To digraph 2,
Delay, τ=0.01s of selection is greater than two the limit of time delay and demonstrates preceding step so closed-loop system is unstable
Four conclusion.
(2) when communication topology figure is non-directed graph Fig. 1, suitable τ, k are selectedP,kDTo guarantee the stability of closed-loop system,
Compare the position tracking error under different parameters.
Selection parameter τ=0.01s, kP=10, kD=12, then ck=14.4 > 13.583, simulation result is as shown in figure 4, every
The position tracking error convergence of a intelligent body, and final boundary is less than 0.004.Compared to a upper sub-step (1), parameter is only changed
kDValue, make condition (7) set up.
Select smaller controller gain kP=1, kD=4, then ck=16 > 13.583 consider three kinds of case propagation delays τ respectively
=0.01s, τ=0.1s, τ=10s, corresponding simulation result such as Fig. 5, shown in 6,7.As long as can be seen that ckMeet condition
(7), even if delay, τ is very big, closed-loop system is also stable, and time delay is smaller, and the final boundary of tracking error is smaller.Particularly,
Prolong τ=0.01s upon selection, the final boundary of tracking error is 0.01 or so, location track value compared to Fig. 5 (a), so small
Tracking error can be ignored.
(3) when communication topology figure is digraph Fig. 2, suitable τ, k are selectedP,kDTo guarantee the stability of closed-loop system,
Compare the position tracking error under different parameters.
Selection parameter τ=0.01s, kP=10, kD=5, then ck=2.5 > 2, simulation result is as shown in figure 8, each intelligence
The position tracking error convergence of body, and final boundary is less than 0.002.Compared to a upper sub-step (1), parameter k is only changedD's
Value sets up condition (7).
Select smaller controller gain kP=1, kD=2, then ck=4 > 2, respectively consider three kinds of case propagation delays τ=
0.01s, τ=0.1s, τ=10s, corresponding simulation result such as Fig. 9, shown in 10,11.If as can be seen that meet condition (7),
Even if delay, τ is very big, closed-loop system is also stable, and time delay is smaller, and the final boundary of tracking error is smaller.Particularly, it is elected to
Delay, τ=0.01s is selected, the final boundary of tracking error is 0.005 or so, location track value compared to Fig. 9 (a), so small
Tracking error can be ignored.
5th step selects different control law parameter τ, k to frame of reference (10)P,kD, each under comparative analysis different situations
The tracking effect of a intelligent body
(1) when communication topology figure is non-directed graph Fig. 1, suitable τ, k are selectedP,kDTo guarantee the stability of closed-loop system,
Compare the position tracking error under different parameters.
Selection parameter kP=1, kD=6, then ck=36 > 13.583, respectively consider two kinds of case propagation delays τ=0.01s, τ=
10s, corresponding simulation result are as shown in figure 12.Under two kinds of case propagation delays, tracking error all converges to 0, because at this timeDemonstrate the conclusion of step 3 in summary of the invention.
(2) when communication topology figure is digraph Fig. 2, suitable τ, k are selectedP,kDTo guarantee the stability of closed-loop system,
Compare the position tracking error under different parameters.
Selection parameter kP=1, kD=4, then ck=16 > 4 consider two kinds of case propagation delays τ=0.01s, τ=10s respectively, right
The simulation result answered is as shown in figure 13.Under two kinds of case propagation delays, tracking error all converges to 0, because at this timeDemonstrate the conclusion of step 3 in summary of the invention.
6th the end of the step
When simulation result shows to select parameter appropriate, distributed AC servo system rule (6) can be such that each intelligent volume tracing last issue hopes
Track, tracking error ultimate boundness or converges to 0, realizes the trajectory synchronization tracing control of multi-agent system.Moreover, for
Not the case where track error does not converge to 0, as long as one sufficiently small time delay of selection, the final boundary that tracking error may be implemented are any
It is small.
1 non-directed graph of table and the corresponding relevant parameter of digraph
Claims (1)
1. a kind of cooperative control method based on PD structure and neighbours' Delay control signal, this method comprises:
Step 1: building tracking error signal, local neighborhood synchronous error signal, communication topology figure；
For the system for having n second order multiple agent:
Wherein pi(t)、qi(t)、ui(t) position, the speed, control input of i-th of intelligent body are respectively indicated,Point
The position of i-th of intelligent body, the derivative of speed are not indicated, i.e.,Indicate the acceleration of i-th of intelligent body；
The traffic model of multi-agent system is established using algebraic graph theory；
For the multi-agent system comprising n follower and 1 pilotage people, regard intelligent body each in system as a point,
Information exchange between intelligent body regards side between points as, with a digraph or non-directed graphIt is more to portray
The Communication topology of multiagent system, whereinFor point set, v1,v2,…,vnFor n point,Side collection；If A=[aij]∈Rn×nFor figureAdjacency matrix, Rn×nIt indicates that n × n ties up real linear space, uses
aijThe connected relation between node is indicated, and if only if (vj,vi) ∈ ε when aij=1, otherwise aij=0, if aii=0；Definition
Indegree matrix D=diag (d1, d2,…dn), whereinNiIndicate the set of node i neighbours；It is asked considering to track
When topic, figure is usedExpression contains the system communication topological diagram of n follower and a pilotage people, uses matrix B=diag
(b1,b2,…bn) come describe the information between pilotage people and follower transmitting situation, if i-th of follower can receive navigator
The information of person, then bi=1, otherwise bi=0；Use μiRepresenting matrix A (D+B)-1Characteristic value, i=1,2 ... n；
The kinetic model of frame of reference are as follows:
Wherein: p0(t), q0(t) desired locations and speed of frame of reference generation are respectively indicated,It respectively indicates
The position of frame of reference, speed derivative, i.e.,Indicate the acceleration of frame of reference, u0(t) input of frame of reference is indicated
Signal；
Tracking error signal are as follows:
Wherein:Indicate the position of i-th of intelligent body and the error of desired locations,Indicate the speed of i-th of intelligent body
With the error of desired speed；
Local neighborhood synchronous error signal:
Wherein:epi(t) position synchronous error is indicated,eqi(t) speed synchronous error is indicated；
The communication topology figure of configuration system, and meet following two condition: 1)There are directed spanning trees；2) matrix A (D+B)-1
N all characteristic value μiIt is all real number；
Step 2: inputting information and local neighborhood synchronous error signal using the control of last time neighbours, design distributed AC servo system
Device；
Control target is that the position of each intelligent body i can track the position of pilotage people while realize between each other same
Step, the neighbours based on intelligent body i control input u thusjWith synchronous error signal epiAnd eqi, design distributed AC servo system rule are as follows:
Whereinui(t) the control input of i-th of intelligent body, u are indicatedj(t) i-th of intelligent body is indicated
Control input, kP> 0 is ratio term coefficient, kD> 0 is differential term coefficient；It is asked in order to avoid there is algebraic loop when control law resolving
Topic controls input u in neighboursjIn introduce fixed delay τ, the improved form for obtaining control law is as follows:
Step 3: when guaranteeing that closed-loop system is stablized, design parameter kPAnd kDThe condition that should meet, to guarantee multi-agent system track
The effect of synchronized tracking is met the requirements；
DefinitionckIt is necessary to meet following condition:
Determine design parameter kPAnd kD, so that condition (7) meets, then to any delay, τ > 0, tracking errorThe stability of uniform ultimate bounded, i.e. multi-agent system is unrelated with delay, τ；In addition, if pilotage people
Acceleration u0(t) be the time a Lipchitz function, then τ is smaller, the final boundary of tracking error is smaller；Particularly,
If u0(t) meet；
As t → ∞, tracking errorAsymptotic convergence is to null vector；
Step 4: if distributed director parameter kP,kDIt is unsatisfactory for the upper bound that calculation delay τ allows when condition (7), when the reality of τ
When value is less than the upper bound, multi-agent system is stablized；Specifically include following sub-step:
(1) μ is calculatedi,With
(2) it solves equationObtain θ；
(3) it calculates
(4) maximum delay that can guarantee system stability is calculated
Step 5: Collaborative Control is carried out to multiple agent according to the result of step 2,3,4.
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